All linear programmes can be represented in standard form

Statement

Lemma

All linear programmes can be represented in standard form.

Proof

Suppose we have a Linear programme of generic form. We need to turn this into a standard form where:

• We are maximising an objective function,
• All constraints use the same inequality, and
• All variables are greater than zero.

If the objective function is this can be converted to a maximum by instead looking at Suppose we have an equality constraint this can be converted into two inequalities Suppose we have greater than or equal to inequality this can be converted into a less than inequality by multiplying through by negative one Lastly suppose we have a variable that can be negative. We can separated it into its positive and negative components where . We can then replace the use of by using the before substitution.

This transformation into a general form may increase the number of variables and number on constraints.